Differential Harnack Inequalities and the Ricci Flow (EMS Series of Lectures in Mathematics)

In 2002, Grisha Perelman presented a new kind of differential Harnack inequality which involves both the (adjoint) linear heat equation and the Ricci flow. This led to a completely new approach to the Ricci flow that allowed interpretation as a gradient flow which maximizes different entropy functionals. The goal of this book is to explain this analytic tool in full detail for the two examples of the linear heat equation and the Ricci flow. It begins with the original Li-Yau result, presents Hamilton's Harnack inequalities for the Ricci flow, and ends with Perelman's entropy formulas and space-time geodesics. The book is a self-contained, modern introduction to the Ricci flow and the analytic methods to study it. It is primarily addressed to students who have a basic introductory knowledge of analysis and of Riemannian geometry and who are attracted to further study in geometric analysis. No previous knowledge of differential Harnack inequalities or the Ricci flow is required. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society. The classical Harnack inequalities play an important role in the study of parabolic partial differential equations. The idea of finding a differential version of such a classical Harnack inequality goes back to Peter Li and Shing Tung Yau, who introduced a pointwise gradient estimate for a solution of the linear heat equation on a manifold which leads to a classical Harnack type inequality if being integrated along a path. Their idea has been successfully adopted and generalized to (nonlinear) geometric heat flows such as mean curvature flow or Ricci flow; most of this work was done by Richard Hamilton. In 2002, Grisha Perelman presented a new kind of differential Harnack inequality which involves both the (adjoint) linear heat equation and the Ricci flow. This led to a completely new approach to the Ricci flow that allowed interpretation as a gradient flow which maximizes different entropy functionals. This approach forms the main analytic core of Perelman's attempt to prove the Poincaré conjecture. It is, however, of completely independent interest and may as well prove useful in various other areas, such as, for instance, the theory of Kähler manifolds. The goal of this book is to explain this analytic tool in full detail for the two examples of the linear heat equation and the Ricci flow. It begins with the original Li-Yau result, presents Hamilton's Harnack inequalities for the Ricci flow, and ends with Perelman's entropy formulas and space-time geodesics. The text is a self-contained, modern introduction to the Ricci flow and the analytic methods to study it. It is primarily addressed to students who have a basic introductory knowledge of analysis and of Riemannian geometry and who are attracted to further study in geometric analysis. No previous knowledge of differential Harnack inequalities or the Ricci flow is required Preface......Page 6 Introduction......Page 10 Riemannian metric and curvature tensors......Page 20 Variation formulas......Page 23 Einstein–Hilbert functional and Ricci flow......Page 26 Evolution equations under Ricci flow......Page 28 Adjoint heat equation and gradient solitons......Page 31 Differential Harnack inequalities......Page 37 The Li–Yau Harnack inequality......Page 39 Hamilton's matrix Harnack inequality......Page 43 Harnack inequalities for the Ricci flow......Page 51 The static case, part I......Page 56 Entropy for steady Ricci solitons......Page 58 The static case, part II......Page 62 Entropy for shrinking solitons......Page 68 Entropy for Ricci expanders......Page 74 Reduced distance and reduced volume......Page 76 The static case......Page 77 Perelman's `39`42`"613A``45`47`"603AL-length and `39`42`"613A``45`47`"603AL-geodesics......Page 82 Monotonicity of the reduced volume......Page 86 Bibliography......Page 96 List of symbols......Page 98 Index......Page 100 Preface Introduction 1. FOUNDATIONAL MATERIAL. Riemannian metric and curvature tensors Variation formulas Einstein-Hilbert functional and Ricci flow Evolution equations under Ricci flow Adjoint heat equation and gradient solitons 2. DIFFERENTIAL HARNACK INEQUALITIES. The Li-Yau Harnack inequality Hamilton's matrix Harnack inequality Harnack inequalities for the Ricci flow 3. ENTROPY FORMULAS. The static case, part I Entropy for steady Ricci solitons The static case, part II Entropy for shrinking solitons Entropy for Ricci expanders 4. REDUCED DISTANCE AND REDUCED VOLUME. The static case Perelman's L-length and L-geodesic Monotonicity of the reduced volume Bibliography List of symbols Index. "The text is a self-contained, modern introduction to the Ricci flow and the analytic methods to study it. It is primarily addressed to students who have a basic introductory knowledge of analysis and of Riemannian geometry and who are attracted to further study in geometric analysis. No previous knowledge of differential Harnack inequalities or the Ricci flow is required."--BOOK JACKET